To answer your question, yes, rosin is a neutral substance, it is a good flux and does not cause corrosion. It makes no difference as to steel or damascus. The only down side is it is very heat-sensitive and chars easily, so tinning has to be dove with a soldering iron, electic, is easiest, about 600 Watts. I have never used a non-electric iron, but I would imagine one heated on an electric hot plate would work, but you would have to keep it up to temperature to melt the solder and and not over-heat to tin the barrels. You need a 600 Watt iron, they are available used on ebay, but make sure it works before it is yours.
Just my opinion, but it is far easier to tin the barrels and ribs using an "active" flux, like Oteys No 5, available in home centers, the solder flows quite easy using it, but you need to remove the salt and tallow that the salt it is mixed with; it is mixed not compounded. Washing the parts with mineral spirits and then washing with a strong detergent like Superclean diluted about 4:1. To demonstrate the need to wash all the flux off after tinning, try tinning a piece if steel and just let it sit over a few days. One of the active salts in some fluxes is a component in some rust blue solutions.
Final assembly is with rosin flux. Taking apart a good quality barrel set often shows no corrosion at all, and the void between the barrels has ben sealed up for years.
And the rosin cleans up with alcohol and steel wool.